This document discusses Bayesian reliability demonstration tests (BRDT) in the design for reliability (DFR) process. It presents challenges with traditional reliability demonstration tests, and how BRDT can help address these challenges by incorporating prior knowledge of a product's reliability from DFR activities. The document outlines how BRDT uses Bayesian statistics with a prior reliability distribution, typically Beta, to calculate posterior reliability and determine confidence levels. It proposes a simplified BRDT algorithm for DFR that constructs the prior reliability distribution based on DFR inputs then performs trade-off studies to determine test parameters like sample size. BRDT allows testing with smaller sample sizes by leveraging reliability information from the DFR process.

1. Bayesian Reliability
Demonstration Test in a
Design for Reliability Process
(可靠性设计过程 – 贝叶斯可
靠性验证试验)
Dr. Mingxiao Jiang
(蒋鸣晓博士)
©2011 ASQ & Presentation Jiang
Presented live on Jul 13th, 2011
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3. Bayesian Reliability Demonstration
Test in a
Design for Reliability Process
Mingxiao Jiang (Medtronic Inc.)
2011
Mingxiao Jiang MEDTRONIC CONFIDENTIAL
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4. Outline
- Design for Reliability (DFR) process
- Challenges of Reliability Demonstration
Test (RDT) in DFR Validation phase
- Bayesian RDT (BRDT) with DFR
- Concluding remarks
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5. Why DFSS and DFR
- Increasing competition
- Increasing product complexity
- Increasing customer expectations of product
performance, quality and reliability
- Decreasing development time
- …
- Higher product quality (“out-of-box” product
performance often quantified by Defective Parts
Per Million) -> DFSS
- Higher product reliability (often as measured by
failure rate, survival function, etc) -> DFR
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6. DFSS vs. DFR
DFSS ANOVA Environmental &
Usage Conditions
DFR
Regression VOC
Life Data Analysis
Flowdown
Physics of Failure
Hypothesis Testing QFD
FMEA Accelerated Life Testing
General Linear Model Control Plans
Reliability Growth
MSA
Sensitivity Analysis Parametric Data Analysis
Modeling
DOE Warranty Predictions
Tolerancing
FA recognition
etc.
etc.
DFR utilizes unique tools to improve reliability.
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7. DFR Process
Development Timeline
Concept, Requirements, Prototype Design
& Prioritization Design Optimization Validation Production
Environment Warranty
Analysis
& Usage Stressors
Reliability Risk Prioritization
DFM & Manufacturing Control Strategy
Requirements & allocation
Prior Products Pareto
Physics of Failure
Stress Testing
FMEA
Parametric Data Analysis
Reliability Demonstration Test
Failure Analysis
Corrective Action & Preventative Action
DFR activities are paced with development.
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8. For Example: Parametric Data Analysis
Few failures
Iceberg Full
distribution
Look at all the parts, not just the few failures!
• Degradation metrics: • Up-stream metrics:
Performance Performance measured
measured during from supplier and during
reliability test manufacturing
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9. Classical Reliability Demonstration Test (CRDT) [1]
r n
  1  R L k R L n  k  1  C
  
k 0  k 
Or 1
RL 
r 1
1 FC ;2r  2;2(n  r )
nr
“Success Run” test (r = 0): RL  (1  C )1 / n
where, n is the test sample size, r is the given allowable
number of failures, C is the confidence level, F( ) is the F
distribution function, and RL is the testing reliability goal.
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10. RDT Challenges in DFR
Sample size, n, needed in RDT:
r=0 C r=2 C
RL 90% 95% 99% RL 90% 95% 99%
90% 22 29 44 90% 52 61 81
95% 45 59 90 95% 105 124 165
99% 230 299 459 99% 531 628 837
r=4 C r=6 C
RL 90% 95% 99% RL 90% 95% 99%
90% 78 89 113 90% 103 116 142
95% 158 181 229 95% 209 234 287
99% 798 913 1157 99% 1051 1182 1452
After reliability allocation in DFR, it is very
challenging to conduct RDT.
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11. RDT: Classical vs. Bayesian
Prior distribution of
RDT planning E.g. Bayesian RDT w/
uniform prior distribution of
 tradeoff:
Reliability
R  one less sample
needed than classical RDT
F C, RL , n, r   0 for zero failure test.
0
0 1
0 Reliability, R 1
• Classical RDT: no prior knowledge of R.
• Bayesian RDT (Ref. 1-5): prior knowledge of R;
challenging math for engineers.
• Bayesian RDT w/ DFR (Ref. 6): prior knowledge of
R weighted more to the right side; math simplified by
spreadsheet calculations.
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12. Bayesian Approach – Discrete Case [1]
Posterior P(Hi is true | data)
Prior P(Hi is true) Conditiona l P(data | H i )

n
 Prior P(Hi is true) Conditiona l P(data | H i )
i 1
Hi (i = 1, …, n) represent a mutually exclusive
exhaustive collection of hypothesis. Suppose that
an event S exists and the conditional probabilities
P(S|Hi) are known. P(Hi) is termed as the prior
probability that Hi is true, and P(Hi|S) is the posterior
probability that Hi is true upon observing S.
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13. Bayesian Approach – Discrete Case, cont’
Example: A large number of identical units are
received from two vendors, A and B. Vendor A
supplies with nine times the number of units that
vendor B supplies. Based on records, defective rate
from A is 2% and defective rate from B is 6%.
Incoming inspection randomly selects one unit and
finds it to be defective. Q: which vendor produced it?
Prior Conditional (Prior P) x Posterior
Vendor probability probability (Conditional P) Probability
A 0.9 0.02 0.018 0.75
B 0.1 0.06 0.006 0.25
1 1
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14. Bayesian Approach – Continuous Case
f ( ) h(S |  )
Prob( | S) 

 f ( ) h(S |  ) d
where, S represents a group of observed
events, θ is a random scalar or vector to
describe the parameters or statistics of the
underline event distribution, Prob(θ|S) is the
posterior probability density function of θ, f(θ) is
the prior probability density function of θ, and
h(S|θ) is the conditional distribution of S.
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15. Bayesian Reliability Demonstration Test (BRDT)
If θ is the reliability R, and S is RDT result, then
f (R ) h (S | R )
Prob(R | S) 
1
0 f (R ) h (S | R ) dR
The confidence level C for the true reliability
within interval [RL, 1] can be obtained as:
1
R L f (R ) h (S | R )dR
C(R L  R  1) 
1
0 f (R ) h (S | R ) dR
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16. h(S|R)
For a certain product with a true reliability R, with
S denoting the outcome of testing the whole
population of sample size n, we have the
conditional probability density function of S given
R:
 n  nr r
h( S | R )    R
r  (1  R)
 
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17. Prior Distribution of Reliability - 1
a
R 1  Rb
Beta distribution: f ( R) 
Bea, b 
a  1  b  1
Where, Bea, b 
a  b  2
Properties of Beta distribution:
- Richness: being able to represent many
states of prior information;
- Conjugation: Beta prior distribution generates
Beta posterior distribution
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18. Prior Distribution of Reliability - 2
a=0, b=0 a=5, b=5 a=10, b=1 a=5, b=0
6
5
4
f(R)
3
2
1
0
0 0.2 0.4 0.6 0.8 1
R
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19. Trade-off: (C, RL, r, n)
1 R n  a  r 1-Rb  r dR
R
L
C ( RL  R  1) 
Be(n  a  r , b  r )
For Success Run test, r = 0:
1 R n  a 1-Rb dR
R
L
C ( RL  R  1) 
Be(n  a, b)
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20. Reliability Prior Distribution in DFR Process - 1
If a product development adopts a DFR process, the prior
distribution of reliability for the components or subsystems to be
validated can be reasonably assumed to be of Beta distribution
being heavily weighted to the right end of (0, 1), with a > b.
20
16 a = 10, b = -1
a = 10, b = 0
a = 10, b = 1
12
Density
a = 10, b = 2
a = 20, b = -1
8 a = 20, b = 0
a = 20, b = 1
a = 20, b = 2
4
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Reliability
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21. Reliability Prior Distribution in DFR Process - 2
• In the DFR risk prioritization phase, the reliability allocated
to a specific component or subsystem could be very high. For
example, a product under development may have an overall
reliability requirement of 90% (for example, first year).
Through FMEA and prior product Pareto assessment, about
10 critical components and subsystems are identified. For the
sake of argument, assuming equal allocation of reliability
requirement to each critical component or subsystem (a much
better allocation approach can be done based on
consideration of cost, risk level, etc) we have approximately
99% reliability as the requirement at one of these individual
components or subsystems.
• Throughout the DFR process with stress testing and PoF
driven corrective actions, the reliability growth is tracked. Of
course, this is subject to RDT to validate.
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22. Bayesian RDT in DFR
Monte Fit prior R
Statistics
Carlo by Beta
of prior R
simulation distribution
Ref:
http://www.barringer1.com/w
dbase.htm;
Construct Telcordia; Simplified
Mil-HDBK-217;
Prior R NSWC (Naval Surface algorithm [6]
Warfare Center) HDBK of
Reliability Prediction
Procedure for Mechanical
Key parameters Equipment (Software Trade-off
identified by MechRel);
study, using
CALCE;
DFR (FMEA, Firm developed; spreadsheet
PoF …) etc
(RL, C, n, r)
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23. Simplified Algorithm for BRDT in DFR
Step 1: Construct a prior reliability:
R P  F( x1, x 2 ,...)
where, RP is the prior reliability, and xk is the
key input variable (could be random) identified
:
in DFR.
Step 2: Obtain the prior distribution of RP:
Monte Carlo simulation results with mean of
prior reliability mRP and variance of prior
reliability VRP
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24. Simplified Algorithm for BRDT in DFR
Step 3: Fit the Beta distribution as the prior
distribution of reliability [1]:
m RP  1  m RP 2 V
RP  m RP  2 
b
V RP :
m RP  b  2  1
a
1  m RP
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25. Simplified Algorithm for BRDT in DFR (Cont’)
Step 4: Conduct the trade-off study among RL,
C, r and n (Ref 6): 100
C   G (k , n, r )
k 0
Where,
k   inta   n  r   1  R
 1   k b r 1
 k  L
G(k , n, r )   
k  b  r  1Beinta   n  r, b  r 
Simple Excel spread sheet calculation; no
programming is needed.
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26. Example
Allocated Reliability goal > 99% @ 5-year
Accelerated RDT w/ usage stress and PoF:
AF = 50  TimeRDT = 0.1yr
Wearout  Weibull shape:  ~ U1, 4
PoF  Weibull scale:  ~ U0.7, 1.4 yr
Zero failure test Confidence
sample size 0.9 0.95 0.99
Classical RDT 230 299 459
Bayesian RDT 81 132 263
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27. Remarks - 1
• Successful application of a Bayesian approach
depends on the prior experience or life data (testing or
field) from previous generations of the product under
design. BRDT can still be used successfully for a totally
new product design and development, based on the
prior distribution characteristics of reliability in a DFR
process.
• DFR activities aid estimation of prior reliability. BRDT
can be integrated into the whole DFR process by linking
it to FMEA, PoF, and reliability requirement flow down or
allocation.
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28. Remarks - 2
• Estimating prior reliability quantifies the interim
effectiveness of the DFR process: the more effective
upstream DFR effort, the more efficient and often
earlier RDT. This can feed into reliability growth
analysis useful for the BRDT design.
• Bayesian reliability approaches involve challenging
mathematical operations for engineers. The illustrated
numerical approach can be used easily by engineers
with any standard spreadsheet calculation
methodology, for success run test or test with failures.
• Bayesian RDT is more efficient and cost effective
than Classical RDT.
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29. References
[1] Kececioglu D, Reliability & Life Testing Handbook, Vol.2, PTR Prentice Hall,
1994.
[2] Kleyner A et al., Bayesian Techniques to Reduce the Sample Size in Automotive
Electronics Attribute Testing, Microelectronics Reliability, Vol. 37, No. 6, 879-883,
1997.
[3] Krolo A et al., Application of Bayes Statistics to Reduce Sample-size Considering
a Lifetime-Ratio, Proceedings of Annual Reliability and Maintainability Symposium,
577-583, 2002.
[4] Lu M-W and Rudy R, Reliability Demonstration Test for a Finite Population,
Quality and Reliability Engineering International, Vol. 17, 33-38, 2001.
[5] Martz H and Waller R, Bayesian Reliability Analysis, Krieger Publishing Company,
1982.
[6] Jiang M and Dummer D, Bayesian Reliability Demonstration Test in a
Design for Reliability Process, PROCEEDINGS Annual Reliability and Maintainability
Symposium, 2009.
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30. Q&A
Thank you!
Mingxiao.Jiang@medtronic.com
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